Lee: If a space is an absolute -Lipschitz retract, with , can you say anything of its injective hull ? Answer: probably no. For instance, the injective hull of Euclidean is which seems very far from although is an absolute -Lipschitz retract.

1. Injective hulls: basics

1.1. Extremal functions

Let be a metric space. Leopoldo Nachbin, in the 1950’s, already considered the set

Definition 1 The minimal elements of are called extremal functions. The set of extremal functions is denoted by .

Note that if is compact, is extremal iff for all , there exists such that . In general,

is extremal iff for all , .

Note that all belong to . This yields a map .

Lemma 2 Extremal functions are -Lipschitz.

Proof: Given , define by and for . Then , so

Let

Then the -norm defines a metric on .

Proposition 3 (Dress) There is a map such that

.

.

Proof: For , set , so is extremal iff . Note that . By definition, for all , ,

so , and . For every ,

Thus

Define (pointwise limit). Then . For all , , so

which tends to , so .

Remark 1 If , an infinite iteration is indeed needed to obtain .

1.2. Injectivity

Proposition 4 is injective.

Proof: Let be metric spaces, let be -Lipschitz. For , , put

Then . and is -Lipschitz. For , . We have

thus . Extend to by setting . Then the extended is -Lipschitz.

Theorem 5 (Isbell)

is injective.

If is -Lipschitz, and fixes pointwise, then is the identity.

If is -Lipschitz and is an isometric embedding, then is an isometric embedding.

If is an isometric embedding and is injective, then there is an isometric embedding such that .

Proof: 2. , so by minimality. 3. follows from 2. and 4. follows from 3.

2. First examples

2.1. Polyhedral structure on , finite

If is bounded, then . If is compact, so is (Arzela-Ascoli).

If is finite, is a polyhedron in . Note that and are convex, but is not in general. The polyhedral structure can be detected by looking at “equality graphs”. For , let be the set of pairs such that . Then is a graph with loops: iff . Also

iff has no isolated vertices.

Say a graph is admissible if there exists such that . Every admissible graph corresponds to a polyhedral cell in , whose interior points are precisely the functions such that . is a face of iff contains .

2.2. Dimension of

If two functions , have the same graph, then for all , , thus

Hence, if there is a path from to in of length , then

On connected components of containing an odd cycle, and coincide. On other (even) components, there is exactly one degree of freedom to vary without changing the graph.

Proposition 6 Define the rank as the number of even components. Then .

Example 1.

Then the possible graphs are an edge (rank one) and one edge with one loop (rank zero). This produces an interval.

Example 2.

Then is a tripod. Indeed, admissible graphs of extremal functions can have at most one even component. The complete graph is admissible but odd, producing a vertex. Two-edge admissible graphs contribute three segments. Hinges with a loop contribute three vertices.

The tripod lies in on a triangular face (standard simplex) at the bottom of polyhedron . It takes infinitely many steps (iterates of ) to map a constant function to . It is a bit stupid.

Lee: why don’t you consider the optimal such that belongs to ?

Example 3 has 4 points with one distance equal to and all other equal to or .

The graph with two disjoint edges (one of length ) is admissible and has rank , it contributes a -cell, a square sitting diagonally in . The resulting polyhedron is the union of the -cell with two opposite protruding edges, each of length .